L(s) = 1 | + 3-s − 2.25·5-s + 3.02·7-s + 9-s − 4.27·11-s + 1.71·13-s − 2.25·15-s + 2.42·17-s + 1.07·19-s + 3.02·21-s − 23-s + 0.0730·25-s + 27-s + 29-s + 9.45·31-s − 4.27·33-s − 6.80·35-s + 7.34·37-s + 1.71·39-s − 3.99·41-s − 6.28·43-s − 2.25·45-s + 0.0164·47-s + 2.13·49-s + 2.42·51-s − 0.890·53-s + 9.62·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.00·5-s + 1.14·7-s + 0.333·9-s − 1.28·11-s + 0.475·13-s − 0.581·15-s + 0.587·17-s + 0.245·19-s + 0.659·21-s − 0.208·23-s + 0.0146·25-s + 0.192·27-s + 0.185·29-s + 1.69·31-s − 0.744·33-s − 1.15·35-s + 1.20·37-s + 0.274·39-s − 0.623·41-s − 0.958·43-s − 0.335·45-s + 0.00239·47-s + 0.305·49-s + 0.339·51-s − 0.122·53-s + 1.29·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.272788196\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.272788196\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 - 3.02T + 7T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 - 1.71T + 13T^{2} \) |
| 17 | \( 1 - 2.42T + 17T^{2} \) |
| 19 | \( 1 - 1.07T + 19T^{2} \) |
| 31 | \( 1 - 9.45T + 31T^{2} \) |
| 37 | \( 1 - 7.34T + 37T^{2} \) |
| 41 | \( 1 + 3.99T + 41T^{2} \) |
| 43 | \( 1 + 6.28T + 43T^{2} \) |
| 47 | \( 1 - 0.0164T + 47T^{2} \) |
| 53 | \( 1 + 0.890T + 53T^{2} \) |
| 59 | \( 1 + 7.48T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 1.64T + 67T^{2} \) |
| 71 | \( 1 - 1.92T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 6.09T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 0.796T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.953535639628357198380382257654, −7.53237024005889941478701889212, −6.56829331661125957856764655243, −5.63197949339279567298658398340, −4.83879833798524886907432814434, −4.36598992950434239194454538522, −3.44949112432146472413271320171, −2.77876057721605905617644008168, −1.81198296217733075029631405198, −0.73578591850459176488466117041,
0.73578591850459176488466117041, 1.81198296217733075029631405198, 2.77876057721605905617644008168, 3.44949112432146472413271320171, 4.36598992950434239194454538522, 4.83879833798524886907432814434, 5.63197949339279567298658398340, 6.56829331661125957856764655243, 7.53237024005889941478701889212, 7.953535639628357198380382257654